Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.


(ii) A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is added to its denominator. Find the fraction.


(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?


(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?


(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

(i) Let x be the fixed charge of the food and y be the charge for food per day. According to the given information,

X + 20y = 1000 .......... (1)


X + 26y = 1180 ........... (2)


Subtracting equation (1) from equation (2), we obtain


6y = 180


Y = 30


Substituting this value in equation (1), we obtain


X + 20 × 30 = 1000


x = 1000 - 600 = 400


x = 400


Hence, fixed charge = Rs 400 And charge per day = Rs 30


(ii) Let the fraction be .


According to the given information,


.......(i)


.......(ii)


Subtracting equation (i) from equation (ii), we obtain


x = 5 ............. (iii)


Putting this value in equation (i), we obtain


15 – y = 3


y = 12


Hence, the fraction is .


(iii) Let the number of right answers and wrong answers be x and y respectively.


According to the given information,


3x –y = 40


4x – 2y = 50 .......(i)


2y – y = 25 .........(ii)


Subtracting equation (ii) from equation (i),


we obtain x = 15 (iii)


Substituting this in equation (ii), we obtain


30 – y = 25


y = 5


Therefore,


number of right answers = 15


And number of wrong answers = 5


Total number of questions = 20


(iv) Let the speed of 1st car and 2nd car be u km/h and v km/h.


Respective speed of both cars while they are travelling in same direction = (u - v) km/h


Respective speed of both cars while they are travelling in opposite directions i.e.,


travelling towards each other = (u + v) km/h


According to the given information,


5(u - v) = 100


= u - v = 20......... (i)


1(u + v) = 100....... (ii)


Adding both the equations, we obtain


2u = 120


u = 60 km/h ...... (iii)


Substituting this value in equation (ii), we obtain v = 40 km/h


Hence, speed of one car = 60 km/h and speed of other car = 40 km/h


(v) Let length and breadth of rectangle be x unit and y unit respectively.


Area = xy


According to the question,


(x - 5) (y + 3) = xy - 9


= 3x – 5y – 6 = 0 ..........(i)


(x + 3) (y + 2) = xy + 67


= 2x + 3y – 61 = 0 ........(ii)


By cross-multiplication method, we obtain




x = 17, y =9


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